Geometry and Topology of Riemannian Manifolds

9971648Grove该项目将处理微分几何及相关领域的全球性问题。特别强调对称和奇异空间在黎曼流形的几何和拓扑中的作用。特别是，该项目背后的指导原则之一是：“对具有正曲率或非负曲率和大等长群的流形进行分类。”这自然包括对具有“旋转对称”的流形的研究，即具有一维轨道空间的流形。本研究还旨在产生具有正曲率或非负曲率的流形的新例子，这是一个众所周知的困难问题。轨道空间通常提供了奇异空间类型的重要例子，这些奇异空间出现在具有低曲率边界的流形的研究中。研究者对奇异空间的研究主要集中在稳定性和逆稳定性问题上。这些问题的答案将有助于更深入地理解黎曼流形的结构。黎曼流形是具有距离函数的弯曲空间的抽象。与三维空间中的曲线和曲面不同，这些散形不一定位于环境空间中，因此它们是抽象空间。这里的研究旨在了解这些流形的各个全局方面。特别地，研究者感兴趣的是这种流形的正或非负曲率如何影响它们的整体性质，有时以微妙和意想不到的方式。其中突出的是各种对称性质，这将是研究者和他的学生研究的一个重要焦点

9971648Grove The project will deal with global problems in differential geometryand related areas. Special emphasis will be devoted to the roles ofsymmetry and of singular spaces in the geometry and topology ofRiemannian manifolds. In particular, one of the guiding principlesbehind the project is: ``Classify manifolds with positive or non-negativecurvature and large isometry groups.'' This naturally includes the studyof manifolds with ``rotational symmetry,'' i.e., manifolds with one-dimensional orbit spaces. This investigation is also aimed at producingnew examples of manifolds with positive or non-negative curvature, aproblem of notorious difficulty. Orbit spaces in general provideimportant examples of the type of singular spaces that arise in the studyof manifolds with lower curvature bounds. The investigator's effortsrelated to singular spaces are focussed on stability and inversestability problems. Answers to such problems will be instrumental in thedeeper understanding of the structure of Riemannian manifolds in general. Riemannian manifolds are an abstraction of curved spaces possessinga distance function. Unlike curves and surfaces in 3-space, thesemanifolds do not necessarily sit inside an ambient space, hence they areabstract spaces. The research here aims to understand various globalaspects of these manifolds. In particular, the investigator is interestedin ways in which the positive or non-negative curvature of such manifoldsaffects their global properties, sometimes in subtle and unexpected ways.Prominent among these are various symmetry properties, which will be animportant focus of the research of the investigator and his student.***